Boundary Value Problems for Functional Differential Equations

296 Pages·1995·7.31 MB·English

by  Henderson J. (ed.)

Boundary Value Problems
for Functional
Differential Equations
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Boundary Value Problems
for Functional
Differential Equations
Editor
Johnny Henderson
College ofS ciences and Mathematics
Auburn University, Alabama, USA
b
l
World Scientific
II
Singapore· New Jersey· London· Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
POBox 128, Farrer Road, Singapore 9128
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS
Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented. without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923, USA.
ISBN: 981-02-2405-2
This book is printed on acid-free paper.
Printed in Singapore by Uto-Print
v
PREFACE
The purpose of Boundary Value Problems for Functional Differential Equa
tions is to present some of the areas of current research in such a way as to be
accessible to a wide audience. In addition, it is hoped that many of these articles can
serve as guides to seminars or additional topics for courses in functional differential
equations.
Functional differential equations have received attention since the 1920's. Within
the development, boundary value problems have played a prominent role in both
the theory and applications dating back to the 1960's. Contributions herein repre
sent not only a flavor of classical results involving, for example, linear methods and
oscillation-nonoscillation techniques, but also modern nonlinear methods for prob
lems involving stability and control as well as cone theoretic, degree theoretic, and
topological transversality strategies. A balance with applications is provided through
a number of papers dealing with a pendulum with dry friction, heat conduction in
a thin stretched resistive wire, problems involving singularities, impulsive systems,
traveling waves, climate modeling, and economic control.
With the importance of boundary value problems for functional differential equa
tions in applications, it is not surprising that as new applications arise, modifications
are required for even the definitions of the basic equations. This was the case for sev
eral researchers who for years conducted a seminar in functional differential equations
at Perm State Technical University. Participants from that seminar have contributed
to this volume. Also, some contributions are devoted to delay Fredholm integral
equations, while a few papers deal with what might be termed as boundary value
problems for delay-difference equations.
I am grateful to my technical typist, Mrs. Rosie Torbert, and to my colleague,
Professor Darrel Hankerson, for their technical assistance in the preparation of this
book. I also express my gratitude to Dr. Anju Goel of World Scientific Publishing.
Johnny Henderson
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vii
CONTENTS
Preface v
Right Focal Point Boundary Value Problems for
Functional-Differential Equations 1
R. P. Agarwal and Q. Sheng
The Ideas and Methods of the Perm Seminar on Boundary
Value Problems 13
N. V. Azbelev
On Extension of the Vallee-Poussin Theorem to Equations
with Aftereffect 23
N. V. Azhelev and L. F. Rakhmatullina
Initial-Boundary Value Problems for Impulsive Parabolic
Functional Differential Equations 37
D. Bainov, Z. Kamont and E. Minchev
Boundary Value Problems on Infinite Intervals 49
J. V. Baxley
Existence of Steady-State Solutions to Some
Constant-Voltage Problems 63
L. E. Bobisud
An Existence Theorem for Hereditary Lagrange Problems on
an Unbounded Interval 73
D. A. Carlson
Dynamical Spectrum for Skew Product Flow in Banach Spaces 85
S. N. Chow and H. Leiva
On Boundary Value Problems for First Order Impulse
Functional Differential Equations 107
A. Domoshnitsky and M. Drakhlin
viii
Linearized Problems and Continuous Dependence 119
J. A. Ehme
Positive Solutions and Conjugate Points for a Class of Linear
Functional Differential Equations 131
P. W. Eloe and J. Henderson
Boundary Value Problems for Second Order Mixed Type
Functional Differential Equations 143
L. H. Erbe. z.-C. Wang and L.-T. Li
The Filippov Approach to Boundary and Initial Value
Problems and Applications 153
R. B. Guenther. J. W. Lee and M. Senkyf{k
An Existence Result for Delay Equations Under Semilinear
Boundary Conditions 165
G. Hetzer
Periodic Solutions of Functional Differential Equations of
Retarded and Neutral Types in Banach Spaces 177
L. H. Hoa aJui K. Schmitt
Existence of Traveling Waves for Reaction Diffusion Equations
of Fisher Type in Periodic Media 187
W. Hudson and B. Zinner
Permanence of Periodic Solutions of Retarded Functional
Differential Equations 201
L. A. da. C. Ladeira and S. M. Tanaka
Method of Quasilinearization for Boundary Value
Problems for Functional Differential Equations 209
V. Lakshmikantham and N. Shahzad
A Resolvent Computation Related to Completeness for
Compact Operators 215
S. M. V. Lunel
The Study of the Solvability of Boundary Value Problems for
Functional Differential Equations: A Constructive Approach 227
V. P. Maksimov and A. N. Rumyantsev
ix
Boundary Value Problems for Neutral Functional
Differential Equations 239
S. K. Ntouyas
Existence Principles for Nonlinear Operator Equations 251
D. O'Regan
Sturmian Theory and Oscillation of a Third Order Linear
Difference Equation 261
A. Peterson
Multipoint Boundary Value Problems for Functional
Differential Equations 269
D. Taunton
Third Order Boundary Value Problems for Differential
Equations with Deviating Arguments 277
P. Ch. Tsamatos
Periodic Solutions and Liapunov Functionals 289
T.-X. Wang
Boundary Value Problems of Second Order Functional
Differential Equations 301
B. Zhang
RIGHT FOCAL POINT BOUNDARY VALUE PROBLEMS FOR
FUNCTIONAL-DIFFERENTIAL EQUATIONS
RAVI P. AGARWAL AND QIN SHENG
Department of Mathematics, National University of Singapore
10 Kent Ridge Crescent, Singapore 0511
Abstract
Inequalities established in 1161 are employed to provide sufficient conditions
for the existence-uniqueness and the convergence of the Picard and the approx-
imate Picard iterative methods for the nth order differential equations with
deviating arguments together with the T-point right focal boundary conditions.
1. Introduction
Let r 2 2 be an integer and -oo < al < . .. < a, < oo be given points. Consider the
nth order differential equation with deviating arguments
x(")(t) = f(t,z 0 ~(t))t ,E [al,~,], (1)
where xow(t) stands for (x(wo,l(t))., . . ,~ (wo,~(o)(.t . ). ),X , (~)(W,,~(,)(~)a)n)d, p(i), 0 5
i 5 q are positive integers. The function f (x,( I))i s assumed to be continuous on
[al,a ,] x RN, where (x) represents (xo,~. ,. . ,X O,~(~.) ., - , x,,,(,)), and N = C:=op (i).
The functions wi,j, 1 5 j 5 p(i), 0 5 i 5 q are continuous on [al,a ,], and wij(t) 5 a,
for all t E [al,a ,]; also, these functions assume the value al at most finite number of
times as t ranges over [al,ar].L et
a = min (a l,
a15t-a,
If a < al, we assume that a function q5 E C(P)[al~l ,is given. For the,given positive
xi=,
integers kl,. -., k , such that k, = n, we define so = 0, sj = xi=, ki. Let r =
min{q, sl - 1). We seek a function x E B = C(')[aa, ,] n C(g)[aa,l l n C(")[ala, ,],
satisfying the r-point right focal conditions:
>
if a < a1 and q sl - 1, then x(')(t) = q5(')(t), 0 5 i 5 q, t E [a,al];
if a < a1 and q < sl - 1, then x(')(t) = q5(;)(t),0 5 i 5 q, t E [a,
+ <
~(~)(a=l )A i.1, q 1 5 i sl - 1; (2)
if a = al, then x(')(al) = Ai,l, 05i<sl-1;